Did anyone mention Smullyan, First-Order Logic, SV 1968? This is where
I first learned about Beth's semantic tableaux, along with Smullyan's
adaption of them as analytic tableaux, in 1977. I applied these ideas
in "A Practical Decision Method for Propositional Dynamic Logic," ACM
STOC'78, pp 326-337. I hadn't seen tableaux used in program
verification before, though that's not to claim they hadn't been.
Certainly in CS as a whole the sequent approach can be traced back to
Prawitz et al and Wang around 1958, who may I guess have been influenced
more by Gentzen than Beth since the latter's book The Foundations of
Mathematics, NH 1959, had not yet appeared. The historians would be a
better judge of this.
Vaughan Pratt